Cutting catheters are known in the art, in which an arterial or endarterectomy catheter has at its distal end, a cutting surface that cuts away fatty deposits. Invariably, however, such catheters cut while being pushed forward. Examples of such catheters are shown in the following: U.S. Pat. Nos. 3,448,741; 4,273,128; and 4,512,344. A disadvantage with such devices is that the continued removal of material in front of the catheter can lead to a build-up of debris that clogs the artery or catheter, so as to render further progress difficult. Another disadvantage is that cutting catheters that cut while advancing, occasionally run into curvatures in the vessel wall that cause the cutting surface to injure the vessel inadvertently. Even if a shield is provided in front of the cutting means, the problem is that when the cutting means cuts towards the shield, it builds up deposits within the catheter, requiring repeated withdrawal for cleaning.